Vectors & 3D Geometry21 questions21 PYQ

Vectors & 3D Geometry — JEE Maths Practice Questions & Solutions

21 questions on Vectors & 3D Geometry with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

mediumPYQ · JEE Main 2026
If the distance of the point (a,2,5)(a,2,5) from the image of (1,2,7)(1,2,7) in the line
x1=y−11=z−22\dfrac{x}{1}=\dfrac{y-1}{1}=\dfrac{z-2}{2}
is 4, then the sum of all possible values of aa is:
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mediumPYQ · JEE Main 2026
A line with direction ratios 1,−1,21,-1,2 intersects the lines x2=y3=z+13\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z+1}{3} and x+1−1=y−21=z4\dfrac{x+1}{-1}=\dfrac{y-2}{1}=\dfrac{z}{4} at the points PP and QQ respectively. If the length of the line segment PQPQ is α\alpha, then 225α2225\alpha^2 is equal to
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mediumPYQ · JEE Main 2026
Let OO be the origin with OP→=a⃗\overrightarrow{OP}=\vec{a} and OQ→=b⃗\overrightarrow{OQ}=\vec{b}. If RR is on OP→\overrightarrow{OP} such that OP→=5OR→\overrightarrow{OP}=5\overrightarrow{OR}, and MM is such that OQ→=5RM→\overrightarrow{OQ}=5\overrightarrow{RM}, then PM→\overrightarrow{PM} equals:
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mediumPYQ · JEE Main 2026
Let △PQR\triangle PQR be such that PP and QQ lie on x+38=y−42=z+12\dfrac{x+3}{8}=\dfrac{y-4}{2}=\dfrac{z+1}{2} and are at distance 6 from R(1,2,3)R(1,2,3). If (α,β,γ)(\alpha,\beta,\gamma) is the centroid of △PQR\triangle PQR, then α+β+γ\alpha+\beta+\gamma equals:
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mediumPYQ · JEE Main 2026
Let the vectors a⃗=−i^+j^+3k^\vec a=-\hat i+\hat j+3\hat k and b⃗=i^+3j^+k^\vec b=\hat i+3\hat j+\hat k. For some λ,μ∈R\lambda,\mu\in\mathbb{R}, let c⃗=λa⃗+μb⃗\vec c=\lambda\vec a+\mu\vec b. If c⃗⋅(3i^−6j^+2k^)=10\vec c\cdot(3\hat i-6\hat j+2\hat k)=10 and c⃗⋅(i^+j^+k^)=−2\vec c\cdot(\hat i+\hat j+\hat k)=-2, then ∣c⃗∣2|\vec c|^2 is equal to
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mediumPYQ · JEE Main 2026
Let a⃗k=(tan⁡θk)i^+j^\vec a_k=(\tan\theta_k)\hat i+\hat j and b⃗k=i^−(cot⁡θk)j^\vec b_k=\hat i-(\cot\theta_k)\hat j, where θk=2k−1π2n+1\theta_k=\dfrac{2^{k-1}\pi}{2^n+1}, for some n∈N, n>5n\in\mathbb{N},\ n>5. Then the value of ∑k=1n∣a⃗k∣2∑k=1n∣b⃗k∣2\dfrac{\displaystyle\sum_{k=1}^{n}|\vec a_k|^2}{\displaystyle\sum_{k=1}^{n}|\vec b_k|^2} is
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mediumPYQ · JEE Main 2026
The square of the distance of the point P(5,6,7)P(5,6,7) from the line x−22=y−53=z−24\dfrac{x-2}{2}=\dfrac{y-5}{3}=\dfrac{z-2}{4} is
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mediumPYQ · JEE Main 2026
The square of the distance of the point of intersection of the lines r⃗=(i^+j^−k^)+λ(ai^−j^)\vec r=(\hat i+\hat j-\hat k)+\lambda(a\hat i-\hat j), a≠0a\ne0, and r⃗=(4i^−k^)+μ(2i^+ak^)\vec r=(4\hat i-\hat k)+\mu(2\hat i+a\hat k) from the origin is
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mediumPYQ · JEE Main 2026
Let a⃗=7 i^+j^−k^\vec a=\sqrt7\,\hat i+\hat j-\hat k and b⃗=j^+2k^\vec b=\hat j+2\hat k. If r⃗\vec r is a vector such that r⃗×a⃗+a⃗×b⃗=0⃗\vec r\times\vec a+\vec a\times\vec b=\vec 0 and r⃗⋅a⃗=0\vec r\cdot\vec a=0, then ∣3r⃗∣2|3\vec r|^2 is equal to
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mediumPYQ · JEE Main 2026
Two adjacent sides of a parallelogram PQRSPQRS are given by PQ⃗=j^+k^\vec{PQ}=\hat j+\hat k and PS⃗=i^−j^\vec{PS}=\hat i-\hat j. If the side PSPS is rotated about the point PP by an acute angle α\alpha in the plane of the parallelogram so that it becomes perpendicular to the side PQPQ, then sin⁡2(5α2)−sin⁡2(α2)\sin^2\left(\dfrac{5\alpha}{2}\right)-\sin^2\left(\dfrac{\alpha}{2}\right) is equal to
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mediumPYQ · JEE Main 2026
Let the point AA be the foot of perpendicular drawn from the point P(a,b,0)P(a,b,0) on the line x−12=y−21=z−α3\dfrac{x-1}{2}=\dfrac{y-2}{1}=\dfrac{z-\alpha}{3}. If the mid point of the line segment PAPA is (0,34,−14)\left(0,\dfrac34,-\dfrac14\right), then the value of a2+b2+α2a^2+b^2+\alpha^2 is equal to
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mediumPYQ · JEE Main 2026
If a⃗=i^+j^+k^\vec a=\hat i+\hat j+\hat k, b⃗=j^−k^\vec b=\hat j-\hat k and c⃗\vec c be three vectors such that a⃗×c⃗=b⃗\vec a\times\vec c=\vec b and a⃗⋅c⃗=3\vec a\cdot\vec c=3, then c⃗⋅(a⃗−2b⃗)\vec c\cdot(\vec a-2\vec b) is equal to
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mediumPYQ · JEE Main 2026
If (2α+1, α2−3α, α−12)\left(2\alpha+1,\ \alpha^2-3\alpha,\ \dfrac{\alpha-1}{2}\right) is the image of (α, 2α, 1)(\alpha,\ 2\alpha,\ 1) in the line x−23=y−12=z1\dfrac{x-2}{3}=\dfrac{y-1}{2}=\dfrac{z}{1}, then the possible value(s) of α\alpha is(are)
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mediumPYQ · JEE Main 2026
Let u^\hat u and v^\hat v be unit vectors inclined at an acute angle such that ∣u^×v^∣=32|\hat u\times\hat v|=\dfrac{\sqrt3}{2}. If A⃗=λu^+v^+(u^×v^)\vec A=\lambda\hat u+\hat v+(\hat u\times\hat v), then λ\lambda is equal to
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mediumPYQ · JEE Main 2026
Let a line LL passing through the point (1,1,1)(1,1,1) be perpendicular to both the vectors 2i^+2j^+k^2\hat i+2\hat j+\hat k and i^+2j^+k^\hat i+2\hat j+\hat k. If P(a,b,c)P(a,b,c) is the foot of perpendicular from the origin on the line LL, then the value of 34(a+b+c)34(a+b+c) is
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mediumPYQ · JEE Main 2026
If the point of intersection of the lines x+13=y+a5=z+b+17\dfrac{x+1}{3}=\dfrac{y+a}{5}=\dfrac{z+b+1}{7} and x−21=y−b4=z−2a7\dfrac{x-2}{1}=\dfrac{y-b}{4}=\dfrac{z-2a}{7} lies on the xyxy-plane, then the value of a+ba+b is
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mediumPYQ · JEE Main 2026
Let a line LL be perpendicular to both the lines L1:x+13=y+35=z+57L_1:\dfrac{x+1}{3}=\dfrac{y+3}{5}=\dfrac{z+5}{7} and L2:x−21=y−44=z−67L_2:\dfrac{x-2}{1}=\dfrac{y-4}{4}=\dfrac{z-6}{7}. If θ\theta is the acute angle between the lines LL and L3:x−8/72=y−4/71=z2L_3:\dfrac{x-8/7}{2}=\dfrac{y-4/7}{1}=\dfrac{z}{2}, then tan⁡θ\tan\theta is equal to
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mediumPYQ · JEE Main 2026
Let the image of the point P(1,6,a)P(1,6,a) in the line L:x1=y−12=z−a+1bL:\dfrac{x}{1}=\dfrac{y-1}{2}=\dfrac{z-a+1}{b}, b>0b>0, be (a3,0,a+c)\left(\dfrac{a}{3},0,a+c\right). If S(α,β,γ)S(\alpha,\beta,\gamma), α>0\alpha>0, is the point on LL such that the distance of SS from the foot of perpendicular from PP on LL is 2142\sqrt{14}, then α+β+γ\alpha+\beta+\gamma is equal to
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easyPYQ · JEE Main 2026
The square of the distance of the point (−2,−8,6)(-2,-8,6) from the line x−11=y−12=z−1\dfrac{x-1}{1}=\dfrac{y-1}{2}=\dfrac{z}{-1} along the line x+51=y+5−1=z2\dfrac{x+5}{1}=\dfrac{y+5}{-1}=\dfrac{z}{2} is equal to
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easyPYQ · JEE Main 2026
The shortest distance between the lines r⃗=(13i^+2j^+83k^)+λ(2i^−5j^+6k^)\vec r=\left(\dfrac13\hat i+2\hat j+\dfrac83\hat k\right)+\lambda\left(2\hat i-5\hat j+6\hat k\right) and r⃗=(−23i^−13k^)+μ(j^−k^)\vec r=\left(-\dfrac23\hat i-\dfrac13\hat k\right)+\mu\left(\hat j-\hat k\right), λ,μ∈R\lambda,\mu\in\mathbb{R}, is
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easyPYQ · JEE Main 2026
If a⃗\vec a and b⃗\vec b are two vectors such that ∣a⃗∣=2|\vec a|=2 and ∣b⃗∣=3|\vec b|=3, then the maximum value of 3∣(3a⃗+2b⃗)∣+4∣(3a⃗−2b⃗)∣3\big|(3\vec a+2\vec b)\big|+4\big|(3\vec a-2\vec b)\big| is
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